Page:Popular Science Monthly Volume 83.djvu/391

Rh distances perpendicular to the $$xy$$ plane are positive if measured above, negative if measured below. This notation enables us to locate any point in our space.

Now we know of $$2$$-space only as a section of $$3$$-space, and a duodim is purely an imaginary being to us; and we know of $$1$$-space only as a section of $$2$$-space (and therefore of $$3$$-space), and the unodim is imaginary. We have seen that a duodim might interfere with life in $$1$$-space, but the unodim would not know at all what had caused the

interference. We have also seen that a tridim might in a similar way interfere with life in $$2$$-space. The important point to observe is that in either case the inhabitant of the lower space would not understand what had caused the change.

A duodim could lock up his treasure in circular or polygonal vaults, such as "$$a$$" or "$$b$$," safe from $$2$$-space intruders, but a tridim could help himself to anything he pleased without breaking the sides of the vault. By analogy, a $$4$$-space being could do many things in $$3$$-space impossible to man and entirely inexplicable to him. No $$3$$-space safe or vault would be secure from a $$4$$-space burglar. He could get a ball out of a hollow shell without breaking the surface, he could get out the

contents of an egg without cracking the shell and enjoy the kernel of a nut without the use of a nut-cracker.

A geometrical illustration similar to those already given is found in Fig. 9. Here "$$a$$" and "$$b$$" are symmetrical tetrahedrons, in length